An extension of Birkhoff’s theorem with an application to determinants Original Research Article

Let image denote the algebra of n×n matrices over the field image of complex, or real, numbers. Given a self-adjoint involution image, that is, J=J*,J2=I, let us consider image endowed with the indefinite inner product [,] induced by J and defined by image. Assuming that (r,n-r), 0less-than-or-equals, slantrless-than-or-equals, slantn, is the inertia of J, without loss of generality we may assume J=diag(j1,cdots, three dots, centered,jn)=Ircircled plus-In-r. For image, the matrices of the form T=(tik2jijk), with all line sums equal to 1, are called J-doubly stochastic matrices. In the particular case rset membership, variant{0,n}, these matrices reduce to doubly stochastic matrices, that is, non-negative real matrices with all line sums equal to 1. A generalization of Birkhoff’s theorem on doubly stochastic matrices is obtained for J-doubly stochastic matrices and an application to determinants is presented.